Although the special points which are located at lower-half \(\omega \) -plane are not related to the informations of quantum chaos, the retarded Green's functions are also not unique at these special points. The general pole-skipping points in the lower-half \(\omega \) -plane are located at negative integer (imaginary) Matsubara frequencies \(\mathfrak {w}_{n}=-in\)  \((n=1,2\dots )\) , where \(\mathfrak {w}=\frac{\omega }{2\pi T}\) . These special points have been found in BTZ black hole [1]}, Schwarzschild-AdS spacetime [2]}, 2D CFT [3]}, a holographic system with chiral anomaly [4]}, a holographic system at finite chemical potential [5]}, hyperbolic space[6]}, the large \(q\)  limit of SYK chain [7]} and anisotropic plasma [8]}. The lower-half \(\omega \) -plane pole-skipping points which are at non-integer values of \(i\mathfrak {w}\)  have be found in 2D CFT [3]} and the large \(q\)  limit of SYK chain [7]}. Although the Green's functions at the non-integer values of \(i\mathfrak {w}\)  pole-skipping points can not be defined uniquely, the incoming boundary condition is unique. The Green's functions at the non-integer values of \(i\mathfrak {w}\)  pole-skipping points are not related to the incoming solution at the horizon [11]}. So if we use the holographic near horizon analysis to calculate the lower half-plane pole-skipping points, we just obtain the negative integer points.
